Listening to Ramstein, "Keine Lust", I heard "Mir ist kalt, so kalt.". I thought it would have been "Ich bin kalt". Did I hear it incorrectly, or is there a rule I don't remember? (It's been 20 years since German class.)

A possible translation of "jemand ist kalt" could be "someone is callous", e.g. a callous killer.

Said that, I try to interpret the Rammstein lyrics. First I will mention that's not easy, because Rammstein songs are often very ambiguous.

Lustlos fasse ich mich an,
und merke bald,
ich bin schon lange kalt.

So kalt,
Mir ist kalt,

Before these lines they sing something about lying in the snow, so most likely "ich bin schon lange kalt" is referring to the temperature of his body. But there could also be a slight connotation regarding the second meaning of "kalt sein", because of "Ich hab keine Lust" which is not only the title of the song ("Keine Lust"), but is also repeated in almost every line. "Keine Lust haben", meaning that you don't bother, could lead into the direction of "kalt sein" in its figurative meaning. Though, reading the lyrics I wouldn't interpret it that way, just wanted to say that, since Rammstein songs are - as I already mentioned - very ambiguous.
Back to content, after he song that his body is cold he continues, by saying "Mir is kalt", that he is feeling cold.

Anyway, just to make it a bit more confusing. In some region instead of "Mir ist kalt", you may here:

While writing this answer, i saw you already found the solution yourself, but maybe some of the links i've collected are still useful.

Nice question - you've hit the difficult problem of sentences without a subject. "Mir ist kalt." lacks a subject in nominative case, but the sentence is still grammatically correct.

The other way - "Ich bin kalt." is correct as well, but they have different meanings: "Mir ist kalt." describes a perception or a feeling - in English: "to feel cold". On the other hand, "Ich bin kalt." just describes a fact: "I'm cold." So, you could say "Das Wasser ist kalt.", but it would sound odd saying "Dem Wasser ist kalt."

There are different descriptions for the grammatical structure in sentences like "Mir ist kalt.":

I should have googled first, I guess, but I did it after I read @efie 's answer. I found this and it explained that "mir ist kalt" is more like saying "it is cold to me" and that this and other phrases involving feeling and impressions often take this reflexive form.

It's okay to answer your own question but the other answer says exactly the same thing.
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GigiliJun 6 '12 at 18:01

I thought I had replied to this comment earlier. Guess i never clicked the button. The other answer that says the same thing was posted about the same time as mine, so I didn't see it until later. I accepted it anyway, because it was more thorough than mine.
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TecBratJun 7 '12 at 2:40

I meant the one by "efie" which was posted two hours before yours.
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GigiliJun 7 '12 at 6:07

No, completely different. The one from @tohuwawohu that you edited explains it better.
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TecBratJun 7 '12 at 13:03

1

Another example of this is "mir ist schlecht" (which means something completely different than "ich bin schlecht") and "mir ist schwindlig". On the other hand, you say "ich bin müde", not "mir ist müde". Also "ich bin hungrig" and "ich bin durstig".
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celtschkJun 12 '12 at 18:07

There may be nuances, regionally. But for me (Southern Germany), "mir ist kalt" sounds perfect as a translation for "I'm cold."

I would go so far as saying "ich bin kalt" is quite uncommon. At the very least, it's the stronger phrase (as in: "about to freeze"). As a second-person phrase, I think "er ist kalt" could even be a euphemism for "he's dead".

When it comes to temperature, you would use "Mir ist..." normally. There are other connotations to using "Ich bin kalt / warm / heiß", which you would want to avoid.

For example,

"Ich bin heiß" would mean "I am horny"

"Ich bin warm" means "I am gay"

"Ich bin kalt" means "I am frigid"

I suppose that "Mir ist kalt" is a statement about an external condition that is affecting a person, as opposed to "I am cold" describing your own internal condition. I think constructions like this can vary because of slight differences in perceptions of things.